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Lecture 2 Rigid-Body Motion and Imaging Geometry

Lecture 2 Rigid-Body Motion and Imaging Geometry. OUTLINE. 3-D EUCLIDEAN SPACE & RIGID-BODY MOTION Coordinates and coordinate frames Rigid-body motion and homogeneous coordinates. GEOMETRIC MODELS OF IMAGE FORMATION Pinhole camera model. CAMERA INTRINSIC PARAMETERS

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Lecture 2 Rigid-Body Motion and Imaging Geometry

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  1. Lecture 2 Rigid-Body Motion and Imaging Geometry Invitation to 3D vision

  2. OUTLINE • 3-D EUCLIDEAN SPACE & RIGID-BODY MOTION • Coordinates and coordinate frames • Rigid-body motion and homogeneous • coordinates • GEOMETRIC MODELS OF IMAGE FORMATION • Pinhole camera model • CAMERA INTRINSIC PARAMETERS • From metric to pixel coordinates SUMMARY OF NOTATION Invitation to 3D vision

  3. Coordinates of a point in space: Standard base vectors: 3-D EUCLIDEAN SPACE - Cartesian Coordinate Frame Invitation to 3D vision

  4. Coordinates of the vector : 3-D EUCLIDEAN SPACE - Vectors A “free” vector is defined by a pair of points : Invitation to 3D vision

  5. Cross product between two vectors: 3-D EUCLIDEAN SPACE – Inner Product and Cross Product Inner product between two vectors: Invitation to 3D vision

  6. Coordinates are related by: RIGID-BODY MOTION – Rotation Rotation matrix: Invitation to 3D vision

  7. Coordinates are related by: Velocities are related by: RIGID-BODY MOTION – Rotation and Translation Invitation to 3D vision

  8. Homogeneous coordinates: Homogeneous coordinates/velocities are related by: RIGID-BODY MOTION – Homogeneous Coordinates 3-D coordinates are related by: Invitation to 3D vision

  9. IMAGE FORMATION – Perspective Imaging “The Scholar of Athens,” Raphael, 1518 Invitation to 3D vision Image courtesy of C. Taylor

  10. Frontal pinhole IMAGE FORMATION – Pinhole Camera Model Pinhole Invitation to 3D vision

  11. Homogeneous coordinates IMAGE FORMATION – Pinhole Camera Model 2-D coordinates Invitation to 3D vision

  12. metric coordinates Linear transformation pixel coordinates CAMERA PARAMETERS – Pixel Coordinates Invitation to 3D vision

  13. Calibration matrix (intrinsic parameters) Projection matrix Camera model CAMERA PARAMETERS – Calibration Matrix and Camera Model Pinhole camera Pixel coordinates Invitation to 3D vision

  14. Projection of a 3-D point to an image plane IMAGE FORMATION – Image of a Point Homogeneous coordinates of a 3-D point Homogeneous coordinates of its 2-D image Invitation to 3D vision

  15. Homogeneous representation of its 2-D image Projection of a 3-D line to an image plane IMAGE FORMATION – Image of a Line Homogeneous representation of a 3-D line Invitation to 3D vision

  16. SUMMARY OF NOTATION – Multiple Images . . . • Images are all “incident” at the corresponding features in space; • Features in space have many types of incidence relationships; • Features in space have many types of metric relationships. Invitation to 3D vision

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